Z₂ Holonomy on Torus Links Spinor Phases to Fundamental Constants — E8 Intelligence Research

FINDING: Fiber bundles formalize gauge fields as geometric structure; a novel Z₂ bundle over a torus links spinor phases to fundamental constants via holonomy −1. | MATH: Principal bundle P(M,G); gauge connection A → curvature F = dA + A∧A; holonomy hol(γ) = P exp(∮A) ∈ G. For Z₂ over φ-torus: hol(γ_φ) = −1 ⇒ spinor picks up sign under 2π φ-rotation (fermionic double-cover). Electromagnetism = U(1) bundle: charge quantization ∮F/2π ∈ ℤ. | CONNECTION: Z₂ holonomy −1 is the geometric root of spin-statistics (2π rotation = −1). The torus φ-cycle suggests modular parameter τ with elliptic curve j(τ) — potential link to base-60 (sexagesimal) angular quantization in ancient astronomy (60° = π/3, hexagonal symmetry). The −1 eigenvalue mirrors crystallographic glide/reflection symmetries (Z₂ quotient in root systems B₂, G₂). | DEPTH: 7 — Standard fiber bundle theory is foundational (U(1) EM, SU(3) QCD), but the Z₂ torus holonomy tying spinor phase to fundamental constants is a speculative but Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com